Noisy Guesses

We consider the problem of guessing a random, finite-alphabet, secret $n$-vector, where the guesses are transmitted via a noisy channel. We provide a single-letter formula for the best achievable exponential growth rate of the $\rho$--th moment of the number of guesses, as a function of $n$. This formula exhibits a fairly clear insight concerning the penalty due to the noise. We describe two different randomized schemes that achieve the optimal guessing exponent. One of them is fully universal in the sense of being independent of source (that governs the vector to be guessed), the channel (that corrupts the guesses), and the moment power $\rho$. Interestingly, it turns out that, in general, the optimal guessing exponent function exhibits a phase transition when it is examined either as a function of the channel parameters, or as a function of $\rho$: as long as the channel is not too distant (in a certain sense to be defined precisely) from the identity channel (i.e., the clean channel), or equivalently, as long $\rho$ is larger than a certain critical value, $\rho_{\mbox{\tiny c}}$, there is no penalty at all in the guessing exponent, compared to the case of noiseless guessing.